Steady-state error & PID control worksheet with answers

Free printable Steady-state error & PID control worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
R(s)+5(s + 5)s²(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 5(s + 5)/(s²(s + 10)). Find the acceleration error constant Ka.

G(s) = 5(s + 5)/(s²(s + 10))
Ka =
2.Easy
R(s)+10(s + 1)s(s + 1)(s + 2)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 10(s + 1)/(s(s + 1)(s + 2)). What is the system type N?

G(s) = 10(s + 1)/(s(s + 1)(s + 2))
N =
3.Medium
r(t) = 10t+Ks(s + 5)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 5)) must follow the ramp r(t) = 10t with a steady-state error of 0.5. Find the gain K.

G(s) = K/(s(s + 5))r(t) = 10tess = 0.5
K =
4.Medium
r(t) = 5t+Ks(s + 1)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 1)) must follow the ramp r(t) = 5t with a steady-state error of 0.02. Find the gain K.

G(s) = K/(s(s + 1))r(t) = 5tess = 0.02
K =
5.Hard
r(t) = 2t+5 + 0.5/s2(s + 1)(s + 10)C(s)−

A PI controller C(s) = 5 + 0.5/s drives the plant G(s) = 2/((s + 1)(s + 10)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 2t.

C(s) = 5 + 0.5/sG(s) = 2/((s + 1)(s + 10))r(t) = 2t
ess =
6.Hard
R(s)+2 + Kd​s10s(s + 3)C(s)−

A PD controller C(s) = 2 + Kd·s drives the plant G(s) = 10/(s(s + 3)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.6.

G(s) = 10/(s(s + 3))Kp = 2ζ = 0.6
Kd = s

Answer key

  1. 1. Ka = 2.5
  2. 2. N = 1
  3. 3. K = 100
  4. 4. K = 250
  5. 5. ess = 20
  6. 6. Kd = 0.2367 s

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